If `lim_(xto1) (1+ax+bx^(2))^((c)/((x-1)))=e^(3)`, then find the value of bc is _________.

Answer

`(3)`

Explanation

`underset(xto1)lim(1+as+bx^(2))^((e)/(x-1))=e^(3)`
or `e^(underset(xto1)lim(1+as+bx^(2)-1)^((e)/(x-1)))=e^(3)`
or `e^(underset(xto1)lim(c(ax+bx^(2)))/(x-1))=e^(3)`
or `underset(xto1)lim(c(ax+bx^(2)))/(x-1)=3`
or `underset(hto0)lim(c"("a(1+h)+b(1+h)^(2)")")/(1+h-1)=3`
or`underset(hto0)lim((ca+b)+(ac+2b)h+bh^(2))/(h)=3`
or`ca+b=0 " and "ac+2b=3`
or `b=3" and "ac=-3`
Also, the form must be `1^(oo)` for which a+b=0, i.e.,a=-3" and "c=1`.